A RIGOROUS PROOF ON CIRCULAR WIRELENGTH FOR HYPERCUBES

We study embeddings of the n-dimensional hypercube into the circuit with 2nvertices.We prove that the circular wirelength attains a minimum by gray coding;that was called the CT conjecture by Chavez and Trapp(Discrete Applied Mathematics,1998).This problem had claimed to be settled by Ching-Jung Guu in her doctoral dissertation“The circular wirelength problem for hypercubes”(University of California,Riverside,1997).Many argue there are gaps in her proof.We eliminate the gaps in her dissertation.

光RP(k)网络上Hypercube通信模式的波长指派算法

波长指派是光网络设计的基本问题,设计波长指派算法是洞察光网络通信能力的基本方法.基于光RP(k)网络,讨论了其波长指派问题. 含有N=2n个节点的Hypercube通信模式,构造了节点间的一种排列次序Xn,并设计了RP(k)网络上的波长指派算法.在构造该算法的过程中,得到了在环网络上实现n维Hypercube通信模式的波长指派算法.这两个算法具有较高的嵌入效率.在RP(k)网络上,实现Hypercube通信模式需要max{2,52n-5/3}个波长.而在环网络上,实现该通信模式需要复用N/3+N/12个波长,比已有算法需要复用N/3+N/4个波长有较大的改进.这两个算法对于光网络的设计具有较大的指导价值.

On Restricted Connectivity and Extra Connectivity of Hypercubes and Folded Hypercubes

Given a graph G and a non-negative integer h, the h-restricted connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whose deletion disconnects G and every remaining component has the minimum degree of vertex at least h; and the h-extra connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, if any, whose deletion disconnects G and every remaining component has order more than h. This paper shows that for the hypercube Qn and the folded hypercube FQn, κ1（Qn）=κ（1）（Qn）=2n-2 for n≥3, κ2（Qn）=3n-5 for n≥4, κ1（FQn）=κ（1）（FQn）=2n for n≥4 and κ（2）（FQn）=4n-4 for n≥8.

Diagnosabilities of exchanged hypercube networks under the pessimistic one-step diagnosis strategy

The exchanged hypercube EH（s, t） （where s ≥ 1 and t ≥ 1） is obtained by systematically reducing links from a regular hypercube Q,＋t＋l. One-step diagnosis of exchanged hypercubes which involves only one testing phase during which processors test each other is discussed. The diagnosabilities of exchanged hypercubes are studied by using the pessimistic one-step diagno- sis strategy under two kinds of diagnosis models： the PMC model and the MM model. The main results presented here are the two proofs that the degree of diagnosability of the EH（s, t） under pessimistic one-step tl/tl fault diagnosis strategy is 2s where I ≤ s ≤ t （respectively, 2t, where 1 ≤ t ≤ s） based on the PMC model and that it is also 2s where 1 ≤ s ≤ t （respectively, 2t, where 1 ≤ t ≤ s） based on the MM＊ model.

Application of the optimal Latin hypercube design and radial basis function network to collaborative optimization

Improving the efficiency of ship optimization is crucial for modem ship design. Compared with traditional methods, multidisciplinary design optimization （MDO） is a more promising approach. For this reason, Collaborative Optimization （CO） is discussed and analyzed in this paper. As one of the most frequently applied MDO methods, CO promotes autonomy of disciplines while providing a coordinating mechanism guaranteeing progress toward an optimum and maintaining interdisciplinary compatibility. However, there are some difficulties in applying the conventional CO method, such as difficulties in choosing an initial point and tremendous computational requirements. For the purpose of overcoming these problems, optimal Latin hypercube design and Radial basis function network were applied to CO. Optimal Latin hypercube design is a modified Latin Hypercube design. Radial basis function network approximates the optimization model, and is updated during the optimization process to improve accuracy. It is shown by examples that the computing efficiency and robustness of this CO method are higher than with the conventional CO method.

PATHS AND CYCLES EMBEDDING ON FAULTY ENHANCED HYPERCUBE NETWORKS

Let Qn,k （n 〉 3, 1 〈 k ≤ n - 1） be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges, fv and fe be the numbers of faulty vertices and faulty edges, respectively. In this paper, we give three main results. First, a fault-free path P[u, v] of length at least 2n - 2fv - 1 （respectively, 2n - 2fv - 2） can be embedded on Qn,k with fv ＋ f≤ n- 1 when dQn,k （u, v） is odd （respectively, dQ,~,k （u, v） is even）. Secondly, an Q,,k is （n - 2） edgefault-free hyper Hamiltonianaceable when n （ 3） and k have the same parity. Lastly, a fault-free cycle of length at least 2n - 2fv can be embedded on Qn,k with f~ 〈 n - 1 and fv＋f≤2n-4.

t/k-fault diagnosis algorithm of n-dimensional hypercube network based on the MM*model

Compared with accurate diagnosis, the system’s selfdiagnosing capability can be greatly increased through the t/kdiagnosis strategy at most k vertexes to be mistakenly identified as faulty under the comparison model, where k is typically a small number. Based on the Preparata, Metze, and Chien(PMC)model, the n-dimensional hypercube network is proved to be t/kdiagnosable. In this paper, based on the Maeng and Malek(MM)*model, a novel t/k-fault diagnosis(1≤k≤4) algorithm of ndimensional hypercube, called t/k-MM*-DIAG, is proposed to isolate all faulty processors within the set of nodes, among which the number of fault-free nodes identified wrongly as faulty is at most k. The time complexity in our algorithm is only O(2~n n~2).

Multiply-twisted Hypercube with Four or Less Dimensions is Vertex-transitive

P Kulasinghe and S Bettayeb showed that any multiply-twisted hypercube withfive or more dimensions is not vertex-transitive. This note shows that any multiply-twistedhypercube with four or less dimensions is vertex-transitive, and that any multiply-twistedhypercube with three or larger dimensions is not edge-transitive.

VERTEX-FAULT-TOLERANT CYCLES EMBEDDING ON ENHANCED HYPERCUBE NETWORKS

In this paper, we study the enhanced hypercube, an attractive variant of the hypercube and obtained by adding some complementary edges from a hypercube, and focus on cycles embedding on the enhanced hypercube with faulty vertices. Let Fu be the set of faulty vertices in the n-dimensional enhanced hypercube Qn,k （n ≥ 3, 1 ≤ k 〈≤n - 1）. When IFvl = 2, we showed that Qn,k - Fv contains a fault-free cycle of every even length from 4 to 2n - 4 where n （n ≥ 3） and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2n - 4, simultaneously, contains a cycle of every odd length from n-k ＋ 2 to 2^n-3 where n （≥ 3） and k have the different parity. Furthermore, when ｜Fv｜ = fv ≤ n - 2, we prove that there exists the longest fault-free cycle, which is of even length 2^n - 2fv whether n （n ≥ 3） and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2^n - 2fv ＋ 1 in Qn,k - Fv where n （≥ 3） and k have the different parity.

(d,m)-DOMINATING NUMBERS OF HYPERCUBE

This paper shows that the (d,m)-dominating number of the m-dimensional hypercube Q m(m≥4) is 2 for any integer d.[FK(W1*1。*2]m2[FK(W1*1。*2]+2≤d≤m.

On the extra edge-connectivity of hypercubes

The classical hypercube structure is a popular topological architecture in parallel computing environments and a large number of variations based on the hypercube were posed in the past three decades. Reliability evaluation of systems is important to the design and maintenance of multiprocessor systems. The h-extra edge-connectivity of graph G（V, E） is a kind of measure for the reliability of interconnection systems, which is defined as the minimum cardinality of a subset of edge set, if any, whose deletion disconnects G and such that every re- maining component has at least h vertices. This paper shows that the h-extra edge-connectivity 2n-1 2n-1 of the hypercube Qn is a constant 2n-1 for 2n-1/3≤ h2n-1, and n ≥ 4, which extends the result of [Bounding the size of the subgraph induced by m vertices and extra edge-connectivity of hypercubes, Discrete Applied Mathematics, 2013, 161（16）： 2753-2757].

Multidisciplinary Design and Optimization of Satellite Launch Vehicle Using Latin Hypercube Design of Experiments

The design of new Satellite Launch Vehicle （SLV） is of interest, especially when a combination of Solid and Liquid Propulsion is included. Proposed is a conceptual design and optimization technique for multistage Low Earth Orbit （LEO） bound SLV comprising of solid and liquid stages with the use of Genetic Algorithm （GA） as global optimizer. Convergence of GA is improved by introducing initial population based on the Design of Experiments （DOE） Technique. Latin Hypercube Sampling （LHS）-DOE is used for its good space filling properties. LHS is a stratified random procedure that provides an efficient way of sampling variables from their multivariate distributions. In SLV design minimum Gross Lift offWeight （GLOW） concept is traditionally being sought. Since the development costs tend to vary as a function of GLOW, this minimum GLOW is considered as a minimum development cost concept. The design approach is meaningful to initial design sizing purpose for its computational efficiency gives a quick insight into the vehicle performance prior to detailed design.

A Time-Space Optimal Parallel Sorting on a Hypercube

In this paper we discuss a parallel sorting algorithm on a hypercube. Its time complexity is O(n logn/p) +O(n). Here, P is the number of processors available and n, the amount of items to be sorted. Take the problem of time-space optimization into consideration, when P≤ O(log n), this algorithm is both timespace optimal and cost optimization. But this means only speedup is O(P) and it is not linear speedup. Therefore, we further discuss relevant parallel efficiency problems.

Restricted-Faults Identification in Folded Hypercubes under the PMC Diagnostic Model

System-level fault identification is a key subject for maintaining the reliability of multiprocessor interconnected systems. This task requires fast and accurate inferences based on big volume of data, and the problem of fault identification in an unstructured graph has been proved to be NP-hard （non-deterministic polynomial-time hard）. In this paper, we adopt the PMC diagnostic model （first proposed by Preparata, Metze, and Chien） as the foundation of point-to-point probing technology, and a system contains only restricted-faults if every of its fault-free units has at least one fault-free neighbor. Under this condition we propose an efficient method of identifying restricted-faults in the folded hypercube, which is a promising alternative to the popular hypercube topology.

A Fault-Handling Method for the Hamiltonian Cycle in the Hypercube Topology

Many routing protocols,such as distance vector and link-state protocols are used for nding the best paths in a network.To nd the path between the source and destination nodes where every node is visited once with no repeats,Hamiltonian and Hypercube routing protocols are often used.Nonetheless,these algorithms are not designed to solve the problem of a node failure,where one or more nodes become faulty.This paper proposes an efcient modied Fault-free Hamiltonian Cycle based on the Hypercube Topology(FHCHT)to perform a connection between nodes when one or more nodes become faulty.FHCHT can be applied in a different environment to transmit data with a high-reliability connection by nding an alternative path between the source and destination nodes when some nodes fail.Moreover,a proposed Hamiltonian Near Cycle(HNC)scheme has been developed and implemented.HNC implementation results indicated that FHCHT produces alternative cycles relatively similar to a Hamiltonian Cycle for the Hypercube,complete,and random graphs.The implementation of the proposed algorithm in a Hypercube achieved a 31%and 76%reduction in cost compared to the complete and random graphs,respectively.

On Embedding of m-Sequential k-ary Trees into Hypercubes

In this paper, we present an algorithm for embedding an m-sequential k-ary tree into its optimal hypercube with dilation at most 2 and prove its correctness.

Boolean Automorphisms of a Hypercube Coincide with the Linear Isometries

Boolean homomorphisms of a hypercube, which correspond to the morphisms in the category of finite Boolean algebras, coincide with the linear isometries of the category of finite binary metric vector spaces.

On Certain Connected Resolving Parameters of Hypercube Networks

Given a graph , a set is a resolving set if for each pair of distinct vertices there is a vertex such that . A resolving set containing a minimum number of vertices is called a minimum resolving set or a basis for . The cardinality of a minimum resolving set is called the resolving number or dimension of and is denoted by . A resolving set is said to be a star resolving set if it induces a star, and a path resolving set if it induces a path. The minimum cardinality of these sets, denoted respectively by and are called the star resolving number and path resolving number. In this paper we investigate these re-solving parameters for the hypercube networks.

A Proof for g-Good-Neighbor Diagnosability of Exchanged Hypercubes

The diagnosability of a multiprocessor system or an interconnection network is an important research topic. The system and an interconnection network have an underlying topology, which is usually presented by a graph. In this paper, we show proof for the g-good-neighbor diagnosability of the exchanged hypercube EH (s,t) under the PMC model and MM* model.

CONNECTIVITY OF CARTESIAN PRODUCT DIGRAPHS AND FAULT-TOLERANT ROUTINGS OF GENERALIZED HYPERCUBE

Abstract In this paper, the problem of fault tolerant routings in fault tolerant networks is considered. A routing in a network assigns to each ordered pair of nodes a fixed path. All communication among nodes must go on this routing. When either a node or a link in a fault tolerant network fails, the communication from one node to another using this faulty element must be sent via one or more intermediate nodes along a sequence of paths determined by this routing. An important and practical problem is how to choose a routing in the network such that intermediate nodes to ensure communication are small for any fault set. Let C d be a directed cycle of order d . In this paper. The author first discusses connectivity of Cartesian product digraphs, then proves that the Cartesian product digraph C d 1 ×C d 2 ×...×C d n (d i≥2,1≤i≤n) has a routing such that at most one intermediate node is needed to ensure transmission of messages among all non faulty nodes so long as the number of faults is less than n . This is a generalization of Dolev et al's result for the n dimensional cube.